Design and fabrication of a GaN HEMT power amplifier based on hidden Markov model for wireless applications

Improvement of power amplifier’s performance is the desired topic in communication systems. There are many efforts are made to provide good input and output matching, high efficiency, sufficient power gain and appropriate output power. This paper presents a power amplifier with optimized input and output matching networks. In the proposed approach, a new structure of the Hidden Markov Model with 20 hidden states is used for modeling the power amplifier. The widths and lengths of the microstrip lines in the input and output matching networks are defined as the parameters that the Hidden Markov Model should optimize. For validating our algorithm, a power amplifier has been realized based on a 10W GaN HEMT with part number CG2H40010F from the Cree corporation. Measurement results have shown a PAE higher than 50%, a Gain of about 14 dB, and input and output return losses lower than -10 dB over the frequency range of 1.8–2.5 GHz. The proposed PA can be used in wireless applications such as radar systems.


Introduction
One of the essential components in the structure of a transmitter system is the Power Amplifier (PA) [1]. Since the power amplifier is located at the final stage of the transmitter, its efficiency influences the system's overall efficiency. PAs are found in the realization of the many microwaves and millimeter-wave systems including radar and antenna systems [2][3][4][5][6][7][8], cellular phones [9][10][11], electronic warfare [12,13], heating [14,15], and also many other applications that highlight the importance of such component. Due to the wide variety of PA applications, from wireless communication handsets to heating and electronic warfare, PA is designed and biased in a suitable class to satisfy the desired parameters.
As the input signal of a PA is large, it typically operates in nonlinear conditions and the output signal has some unavoidable distortions. On the other hand, the biasing of a PA also affects its non-linear behavior. Generally, class A or class B structures achieve high linearity. However, for obtaining high linearity, PA must be operated in a low-efficiency region and vice versa [16][17][18]. a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 For improving the efficiency of a PA, several techniques are proposed [19]. Using switching power amplifiers and Doherty power amplifiers [20] are two traditional techniques for achieving this goal. The Envelope Tracking (ET) technique improves efficiency by adjusting the supply voltage [21]. Also for improving the linearity and efficiency simultaneously, the outphasing technique can be used [22].
In addition, to apply conventional methods to increase the efficiency of a circuit, the use of innovative and evolutionary methods for optimizing discrete and Monolithic Microwave Integrated Circuits (MMIC) has also become common. Evolutionary algorithms and multi-objective optimization were considered in [23]. Power amplifier optimization based on a nonlinear programming technique was studied in [24]. Particle Swarm Optimization (PSO) was used for the optimization of various PAs in [16,[25][26][27]. Also, Artificial Bee Colony (ABC) and PSO algorithms were applied by Bipin and Rao for the linearization of a PA in [28]. Bayesian optimization for designing broadband and high-efficiency PA was studied in [29], that the proposed algorithm optimized the drain waveforms by maximizing the fundamental output power over the frequency range of 1.5 to 2.5 GHz. Improving the efficiency and gain with automated deep neural learning was obtained over the frequency range of 1.8-2.2 GHz by Koushalvandi et. all in [30].
In this paper, Hidden Markov Model (HMM) is used for improving the parameters of a PA. HMM was first introduced by Baum and Petrie in 1960 [31,32]. HMM is a statistical model that was used to predict events and model the sequences [33,34]. In the proposed approach, the widths and lengths of micro-strips are optimized by HMM. HMM is a robust algorithm for simulating the sequences and comparing them with various lengths. It can be considered as a machine with hidden states whose transition among them causes generating the observable states [35]. In the proposed approach for improving the efficiency, HMM is used for modeling the PA and predicting an optimized sentence that includes the widths and lengths of the microstrips in the RF paths.
The paper is organized as follows: In Sect. 2, the overall structure of HMM is discussed. In Sect. 3, the proposed method and its application in the high-efficiency PA are explained. Measurement results and discussion are presented in Sect. 4 and finally, a conclusion about the proposed approach is presented in Sect. 5.

Hidden Markov model
HMM consists of several hidden states and several observable states. The transitions between hidden states are determined through probability functions represented by matrix elements {a ij } [36,37]. Fig 1 represents a simple example of HMM that contains two hidden states (π 1 and π 2 ), and two observable states (V 1 and V 2 ). In this structure, each of the two hidden states can emit two observable states.
If we have an HMM with n hidden states and m observables states, we can use the following equations for modeling the behavior of HMM [37].
from a hidden state π i to a hidden state π j . Eq 3 states that the sum of the transition probabilities taken out of each hidden state is equal to one. In each hidden state π i , the observable states can be produced [37]. These probabilities are named with emission probabilities that are shown with the elements of matrix Where in Eq 4, v is the set of observable states, and in Eq 5, b jk is the probability of emitting an observable state v k in the hidden state π j . Eq 6 states that the sum of the emission probabilities taken out of each hidden state is equal to one. Initial probabilities determine the starting

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model with each hidden state at the finish. Eq 7 describes the initial probabilities.
Where in Eq 7, π i is the probability of starting the model with a hidden state π(1). As a result, each HMM is summarized with triple λ = (A, B, π) that can model the optimized sequence of widths and lengths of microstrip lines used in the PA matching network.

High-efficiency PA with HMM algorithm
As said, HMM is a robust algorithm for simulating the sequences, so in the proposed PA, we used HMM for modeling the PA and predicting an optimized sentence that includes the widths and lengths of the microstrips in the RF paths. For using HMM, we should first determine the set of hidden states and transition probabilities matrix and the set of observable states and emission probabilities matrix. After determining the overall structure of HMM, we define the sequence that should be optimized from the HMM structure. This sequence is the widths and lengths of the microstrips line in the RF path. Eq 8 shows this sequence.
The number of parameters in Eq 8 is proportional to the number of the microstrips lines in the input and output matching networks. In the proposed PA, 4 lines in the input and 6 lines in the output matching network were used to realize wideband matching. The number of lines for matching the input and output can be increased by the cost of increasing the total area of PA. The proposed matching circuit provides a suitable bandwidth, high efficiency, and a suitable gain while occupying a relatively small area.
The proposed structure of HMM is illustrated in Fig 2. As shown in Fig 2, we considered 20 hidden states for the proposed HMM for modeling 20 parameters in the sequence p described in Eq 8. Each hidden state is the length or width of each microstrip that should be optimized and can generate eight observable states. These eight observable values for each hidden state, are close to the initial values obtained from the loadpull analysis and the design of the corresponding matching networks taking into account the initial tunes.
The proposed structure of As is shown in Eq 9 and Fig 2, in transition probability matrix A, the probability of going from each hidden state to other hidden states is equal to 1. Eq 11 shows the initial optimized values of microstrip lines that are obtained based on the initial tunes of PA from the load-pull simulation and optimization of the input and output matching network. (All values are in millimeters). The set of observable states is determined based on the initial values vectors, p*, in Eq 11. Therefore, for obtaining the emission probabilities matrix, B, at first, we should select eight values (observable states) for each of the initial values (hidden states) in Eq 8. These eight observable values are selected according to the initial values of each of the micro-strips values obtained in Eq 11. The emission probabilities matrix, B, is expressed by Eq 12.
The training of HMM is performed with the Baum-Welch algorithm which is a traditional algorithm for the training of HMM [37]. Eq 13 is the condition related to the sum of emission probabilities taken out of each hidden state, controlled in each training algorithm iteration. All the eight observable values that 20 hidden states can generate are shown in Eq 14 (All values are in millimeters).
In the Baum-Welch algorithm, the Parameters of HMM, are obtained based on the training sequences. The number of training sequences is 160 specified based on Eq 14 (160 = 20*8). For training of HMM, the maximum likelihood concept is used which is defined by Eq 15 [37]: Where in Eq 15, F is the maximum posterior probability of generating sequence S by λ = (A, B,  π). We want to find the sequence of observable states that optimize the PA performance. For obtaining the solution, first, we should define a fitness function. In other words, we want to find that sequence of observable states that minimize our fitness function. The fitness function that we define, is the PAE of PA. Eq 16, defines this fitness function [38].
Where in Eq 16, P DC is the supply power, and P in and P o are the input and output power, respectively. We can consider the P DC as the following equation [1].
In Eq 17, P diss is the dissipation power in PA and P out,f and S 1 n¼2 P out;nf are the fundamental output power and the sum of output powers in the harmonic frequencies, respectively. At the finish, we can state the PAE by Eq 18.
In Eq 18, for optimizing the PAE, we should minimize dissipation power and the sum of output power in the harmonic frequencies. It means for obtaining a maximum PAE, this equation should be minimized: So, Eq 19 is the primary cost function for optimizing the HMM. The algorithm is terminated when the maximum number of iterations is reached or the specified error is below a given threshold value.

Measurement results and discussion
The proposed algorithm used the ADS and Matlab software for the implementation of the Harmonic Balance of PA and HMM, respectively and during the optimization algorithm, ADS is linked to Matlab [39]. For the realization of PA, we use a GaN HEMT with the part number CGH40010F, from the Cree corporation. The substrate specifications are Rogers 4003, with a thickness of 32 mils and ε r = 3.55. It should be noted that a deep AB class is selected for biasing of the transistor. The bias voltage of the drain, V DD is 28V, the bias voltage of the gate, V GG is -2.74 V, and the drain current, I D is 160 mA. In the proposed circuit in Fig 3,    L g2 are fixed and don't vary among the optimization algorithm. After optimizing the PA, the Mu factor of the PA was obtained, as shown in Fig 5. For a PA to be stable, the Mu factor must be greater than one [19]. The simulation results show that the amplifier is stable at all frequencies. Pout, PAE, and Gain versus Pin at the frequency of 2.2 GHz are shown in Fig 9. As shown in Fig 9, the fabricated PA provides a PAE above 61.6%, a power gain above the 14.5 dB, and a Pout above 39.5 dBm in the saturation region, where the input power is between 24 dBm and 30 dBm.  For verifying the proposed algorithm, the PA is optimized by the ADS gradient optimizer. P out , PAE, and Gain are defined as the main goals. Fig 12 shows Table 2, the comparison is considered in the saturation region, where the input power is between 24dBm and 30dBm. The obtained results are also compared with some previously published similar works that are shown in Table 3. As shown in Table 3, the fabricated PA has a better performance in comparison with other similar works. This improvement can be seen in the PAE, Gain, and output power of PA.

Conclusion
In this paper, we designed and fabricated an optimized power amplifier with a GaN HEMT for the wireless application that its widths and lengths were predicted and modeled by HMM. For  doing this, we defined a new structure of HMM for modeling the PA that consists of several hidden and observable states based on the number of microstrip lines in the input and output matching network. The proposed HMM consisted of 20 hidden states and each hidden state emitted 8 observable states so that their values was close to the initial values obtained from the load-pull analysis and the design of the corresponding matching networks taking into account the initial tunes. The maximum likelihood concept was applied for the training of HMM and the sum of the dissipation power and output powers in the harmonic frequencies was defined as a fitness function. After training HMM, we obtained the optimum values of the widths and lengths of the microstrip lines. With the optimized values, we simulated and realized the PA. Also, In all steps of optimization and simulation of the PA, the precise non-linear model of the transistor was used. Measurement results showed that the PA obtained a PAE higher than 50%, a Gain of about 14 dB, and input and output return losses lower than -10 dB over the frequency range of 1.8-2.5 GHz.